A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.

Wednesday, March 21, 2007

Monday in class an excellent question came up: someone (I think it was Tomassino) asked if permutations behave like combinations in the following fashion: "is it true that P(n,k) is the same as P(n,n-k)?"

"I don't know," said. "Let's find out. A minute or so later, we'd completed the computations. Of course, it was little more than three or four lines of simple arithmetic, but the lesson learned (I hope!) was more than simply how to manipulate a few factorials. Rather, "I want you all to know that the authority to do mathematics, to ask questions and to solve them, to prove things, to come up with new theorems and new theories, does not inhere in me. It doesn't lie in your textbook, it doesn't lie in the 'experts,' whoever they are. The authority lies in the mathematics itself, and therefore in anyone who takes the time to learn the mathematics. It lies in the logically sound arguments and valid computations of which mathematics is built. Anyone who can learn the rules of logic and algebra and adhere to them correctly and consistently has authority to do mathematics, and so to ask questions, to answer them, to create new mathematical ideas. Anyone. The authority is in you, if you take the time."

As much as I despise the term (primarily for its blatant capitalist and patriarchalist overtones), "ownership of" the material, or better yet, "partnership with," the material, is an end towards which I hope I help my students strive.

The math ain't mine. It ain't the domain of the experts, the pointy-heads, the mathematical gurus that rest on high in chaired positions in Harvard and Berkeley. Hell, it ain't even theirs.